11/18/2008115-251Great Theoretical Ideas in Computer ScienceIdeas from the courseInductionNumbersRepresentationFinite Counting and ProbabilityA hint of the infiniteInfinite row of dominoesInfinite sums (formal power series)Infinite choice trees, and infinite probability Infinite RAM ModelPlatonic Version:One memory location for each natural number 0, 1, 2, …Aristotelian Version:Whenever you run out of memory, the computer contacts the factory. A maintenance person is flown by helicopter and attaches 1000 Gig of RAM and all programs resume their computations, as if they had never been interrupted.The Ideal Computer:no bound on amount of memoryno bound on amount of timeIdeal Computer is defined as a computer with infinite RAM. You can run a Java program and never have any overflow, or out of memory errors.An Ideal ComputerIt can be programmed to print out:2: 2.0000000000000000000000…1/3: 0.33333333333333333333…φ: 1.6180339887498948482045…e: 2.7182818284559045235336…π: 3.14159265358979323846264…Printing Out An Infinite Sequence..A program P prints out the infinite sequence s0, s1, s2, …, sk, …if when P is executed on an ideal computer, it outputs a sequence of symbols such that-The kthsymbol that it outputs is sk-For every k∈N, P eventually outputs the kthsymbol. I.e., the delay between symbol k and symbol k+1 is not infinite.11/18/20082Computable Real NumbersA real number R is computable if there is a program that prints out the decimal representation of R from left to right. Thus, each digit of R will eventually be output.Are all real numbers computable?Describable NumbersA real number R is describable if it can be denoted unambiguously by a finite piece of English text.2: “Two.”π: “The area of a circle of radius one.”Are all real numbers describable?Is every computable real number, also a describable real number?And what about the other way?Computable R: some program outputs RDescribable R: some sentence denotes RComputable ⇒ describableTheorem:Every computable real is also describableComputable ⇒ describableTheorem:Every computable real is also describableProof: Let R be a computable real that is output by a program P. The following is an unambiguousdescription of R:“The real number output by the following program:” PMORAL: A computer program can be viewed as a description of its output.Syntax: The text of the programSemantics: The real number output by P11/18/20083Are all reals describable?Are all reals computable?We saw thatcomputable ⇒describable, but do we also havedescribable ⇒computable?Questions we will answer in this (and next) lecture…Correspondence PrincipleIf two finite sets can be placed into 1-1 onto correspondence, then they have the same size.Correspondence DefinitionIn fact, we can use the correspondence as the definition: Two finite sets are defined to have the same size if and only if they can be placed into 1-1 onto correspondence.Georg Cantor (1845-1918)Cantor’s Definition (1874)Two sets are defined to have the same size if and only if they can be placed into 1-1 onto correspondence.Cantor’s Definition (1874)Two sets are defined to have the same cardinality if and only if they can be placed into 1-1 onto correspondence.11/18/20084Do N and E have the same cardinality?N = { 0, 1, 2, 3, 4, 5, 6, 7, … }E = { 0, 2, 4, 6, 8, 10, 12, … }The even, natural numbers.E and N do not have the same cardinality! E is a proper subset of N with plenty left over. The attempted correspondence f(x)=x does not take EontoN.E and N do have the same cardinality!N = 0, 1, 2, 3, 4, 5, … E = 0, 2, 4, 6, 8,10, …f(x) = 2x is 1-1 onto. Lesson: Cantor’s definition only requires that some1-1 correspondence between the two sets is onto, not that all 1-1 correspondences are onto. This distinction never arises when the sets are finite.Cantor’s Definition (1874)Two sets are defined to have the same size if and only if they can beplaced into 1-1 onto correspondence.You just have to get used to this slight subtlety in order to argue about infinite sets!11/18/20085Do N and Z have the same cardinality?N = { 0, 1, 2, 3, 4, 5, 6, 7, … }Z = { …, -2, -1, 0, 1, 2, 3, … }No way! Z is infinite in two ways: from 0 to positive infinity and from 0 to negative infinity. Therefore, there are far more integers than naturals.Actually, no!N and Z do have the samecardinality!N = 0, 1, 2, 3, 4, 5, 6 …Z = 0, 1, -1, 2, -2, 3, -3, ….f(x) = x/2 if x is odd-x/2 if x is evenTransitivity LemmaTransitivity LemmaLemma: If f: A→B is 1-1 onto, and g: B→C is 1-1 onto.Then h(x) = g(f(x)) defines a functionh: A→C that is 1-1 ontoHence, N, E, and Z all have the same cardinality.Do N and Q have the same cardinality?N= { 0, 1, 2, 3, 4, 5, 6, 7, …. }Q = The Rational Numbers11/18/20086No way!The rationals are dense: between any two there is a third. You can’t list them one by one without leaving out an infinite number of them.Don’t jump to conclusions!There is a clever way to list the rationals, one at a time, without missing a single one!First, let’s warm up with another interesting example:N can be paired with N×NTheorem: N and N×N have the same cardinalityTheorem: N and N×N have the same cardinality0 1 2 3 4 ……43210The point (x,y)represents the ordered pair (x,y)Theorem: N and N×N have the same cardinality0 1 2 3 4 ……432100123456789The point (x,y)represents the ordered pair (x,y)11/18/20087Defining 1-1 onto f: N -> N×N let i := 0; //will range over Nfor (sum = 0 to forever) {//generate all pairs with this sumfor (x = 0 to sum) {y := sum-xdefine f(i) := the point (x,y)i++;}} Onto the Rationals!The point at x,y represents x/y The point at x,y represents x/y320 1Cantor’s 1877 letter to Dedekind:“I see it, but I don't believe it!”Countable SetsWe call a set countable if it can be placed into 1-1 onto correspondence with the natural numbers N.HenceN, E, Q and Z are all countable.11/18/20088Do N and R have the same cardinality?N = { 0, 1, 2, 3, 4, 5, 6, 7, … }R = The Real NumbersNo way!You will run out of natural numbers long before you match up every real.Now hang on a minute!You can’t be sure that there isn’t some clever correspondence that you haven’t thought of yet.I am sure!Cantor proved it.To do this, he invented a very important technique called“Diagonalization”Theorem: The set R[0,1]of reals between 0 and 1 is not countable.Proof: (by contradiction)Suppose R[0,1]is countable. Let f be a 1-1
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